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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
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0 replies
jlacosta
Mar 2, 2025
0 replies
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k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
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k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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Inequalities
sqing   4
N 19 minutes ago by sqing
Let $ a,b> 0$ and $ a+b=1 . $ Prove that
$$ \frac{1}{a}+\frac{1}{b}\geq \frac{4+\frac{k}{4096}}{1+ ka^7b^7}$$Where $\frac{8192}{3}\geq k>0 .$
$$ \frac{1}{a}+\frac{1}{b}\geq \frac{\frac{14}{3}}{1+ \frac{8192}{3}a^7b^7}$$
4 replies
sqing
Saturday at 3:14 PM
sqing
19 minutes ago
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Bdt quen thuộc
chunchun.math.2010   0
22 minutes ago
Cho a,b,c>=0 .cmr
a^2b+b^2c+c^2a>=ab^2+bc^2+ca^2
0 replies
chunchun.math.2010
22 minutes ago
0 replies
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how to name the numbers on unit circle
Miranda2829   0
25 minutes ago
I am bit confused on how do you call the numbers on unit circle

sin of 30 is 1/2 cosine of 30 is square root 3/2, etc...

is this call trig ratio?, trig functions? or coordinate?

thanks
0 replies
Miranda2829
25 minutes ago
0 replies
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digit reversing and divisibility
roundtablepizza   3
N an hour ago by aidan0626
an interesting problem i thought of:

for what integers k will the following statement be true: if k divides a number, then it will also divide that number reversed.

for example, since 3 divides 321, it also divides 123.

i know this applies for 3, 9, and 11(maybe??) but are there infinitely many more values of k?
3 replies
roundtablepizza
2 hours ago
aidan0626
an hour ago
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Something nice
KhuongTrang   20
N 2 hours ago by KhuongTrang
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
20 replies
KhuongTrang
Nov 1, 2023
KhuongTrang
2 hours ago
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Yet another OH-symmetry
Tintarn   6
N 3 hours ago by drmzjoseph
Source: All-Russian MO 2024 11.6
Let $ABC$ be an acute non-isosceles triangle with circumcircle $\omega$, circumcenter $O$ and orthocenter $H$. We draw a line perpendicular to $AH$ through $O$ and a line perpendicular to $AO$ through $H$. Prove that the points of intersection of these lines with sides $AB$ and $AC$ lie on a circle, which is tangent to $\omega$.
Proposed by A. Kuznetsov
6 replies
1 viewing
Tintarn
Apr 22, 2024
drmzjoseph
3 hours ago
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prove that a chord is tangent to the incircle
ihategeo_1969   2
N 3 hours ago by cursed_tangent1434
Source: SORY 2019 P6
Let $ABC$ be a triangle with incenter $I$ and intouch triangle $DEF$. Let $P$ be the foot of the perpendicular from $D$ onto $EF$. Assume that $BP$, $CP$ intersect the sides $AC$, $AB$ in $Y,Z$ respectively. Finally, let the rays $IP$, $YZ$ meet the circumcircle of $\triangle ABC$ in $R$, $X$ respectively. Prove that the tangent from $X$ to the incircle and the line $RD$ meet on the circumcircle of $\triangle ABC$.

Proposed by Aditya Khurmi
2 replies
ihategeo_1969
Yesterday at 9:01 PM
cursed_tangent1434
3 hours ago
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"A perfect AIME problem"
XAN4   1
N 4 hours ago by maromex
Source: own
Here is a compilcated problem of calculation. I'd really like to know how you solve it.
Find the minimum $n\in\mathbb Z^+$ such that there exists exactly $n$ different functions $f$ such that $f:[1,5]\rightarrow[1,5]$ satisfying $f^n(x)\geq x$.
1 reply
XAN4
Yesterday at 1:05 PM
maromex
4 hours ago
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Cyclic ine
m4thbl3nd3r   9
N 4 hours ago by Nguyenhuyen_AG
Let $a,b,c>0$ such that $a+b+c=3$. Prove that $$a^3b+b^3c+c^3a+9abc\le 12$$May everyone not to upload solutions on this problem anymore until May 15/2025, this is an active problem on Mathematical Reflection! (Thank you Victoria_Discalceata1)
9 replies
m4thbl3nd3r
Saturday at 3:17 PM
Nguyenhuyen_AG
4 hours ago
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Prove that the lines AM,DN,XY are concurrent
orl   75
N 5 hours ago by golden_star_123
Source: IMO 1995, Problem 1, Day 1, IMO Shortlist 1995, G1
Let $ A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $ AC$ and $ BD$ intersect at $ X$ and $ Y$. The line $ XY$ meets $ BC$ at $ Z$. Let $ P$ be a point on the line $ XY$ other than $ Z$. The line $ CP$ intersects the circle with diameter $ AC$ at $ C$ and $ M$, and the line $ BP$ intersects the circle with diameter $ BD$ at $ B$ and $ N$. Prove that the lines $ AM,DN,XY$ are concurrent.
75 replies
orl
Nov 9, 2005
golden_star_123
5 hours ago
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Nice problem
hanzo.ei   8
N 5 hours ago by maromex
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that
\[ f(xy) = f(x)f(y) \;-\; f(x + y) \;+\; 1, \quad \forall x, y \in \mathbb{R}. \]
8 replies
hanzo.ei
Yesterday at 4:31 PM
maromex
5 hours ago
J
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Number of modular sequences with different residues
PerfectPlayer   2
N 5 hours ago by AnSoLiN
Source: Turkey TST 2025 Day 3 P9
Let \(n\) be a positive integer. For every positive integer $1 \leq k \leq n$ the sequence ${\displaystyle {\{ a_{i}+ki\}}_{i=1}^{n }}$ is defined, where $a_1,a_2, \dots ,a_n$ are integers. Among these \(n\) sequences, for at most how many of them does all the elements of the sequence give different remainders when divided by \(n\)?
2 replies
1 viewing
PerfectPlayer
Mar 18, 2025
AnSoLiN
5 hours ago
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Maximizing score of permutations
navi_09220114   4
N 6 hours ago by Manteca
Source: Malaysian IMO TST 2023 P2
Let $a_1, a_2, \cdots, a_n$ be a sequence of real numbers with $a_1+a_2+\cdots+a_n=0$. Define the score $S(\sigma)$ of a permutation $\sigma=(b_1, \cdots b_n)$ of $(a_1, \cdots a_n)$ to be the minima of the sum $$(x_1-b_1)^2+\cdots+(x_n-b_n)^2$$over all real numbers $x_1\le \cdots \le x_n$.

Prove that $S(\sigma)$ attains the maxima over all permutations $\sigma$, if and only if for all $1\le k\le n$, $$b_1+b_2+\cdots+b_k\ge 0.$$
Proposed by Anzo Teh Zhao Yang
4 replies
navi_09220114
Apr 29, 2023
Manteca
6 hours ago
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not all sufficiently large integers are clean
ABCDE   26
N Yesterday at 10:22 PM by mathfun07
Source: 2015 IMO Shortlist C6, Original 2015 IMO #6
Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is clean if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.
26 replies
ABCDE
Jul 7, 2016
mathfun07
Yesterday at 10:22 PM
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Number Theory
Foxellar   3
N Saturday at 3:35 PM by rchokler
Evaluate the following expression:

\[ 19^{17^{15^{13^{11^{9^{7^{5^3}}}}}}} \mod 100 \]
3 replies
Foxellar
Saturday at 11:43 AM
rchokler
Saturday at 3:35 PM
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Number Theory
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Reveal topic tags
number theory
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Foxellar
4 posts
Foxellar
#1 h  Saturday at 11:43 AM
Y by
Evaluate the following expression:

\[ 19^{17^{15^{13^{11^{9^{7^{5^3}}}}}}} \mod 100 \]
This post has been edited 1 time. Last edited by Foxellar, Saturday at 11:45 AM
Reason: Unnecessary
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cube4320
3 posts
cube4320
#2 h  Saturday at 1:59 PM • 1 Y
Y by Foxellar
Solution: Click to reveal hidden text
(this is my first post, sorry if the answer is a bit unclear)
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Mathematicalprodigy37
5 posts
Mathematicalprodigy37
#3 h  Saturday at 2:57 PM • 1 Y
Y by Foxellar
First, use Euler's totient theorem. phi(100)=40. We now need 17^15^...^3 mod 40. phi(40)=16. Now, we need to find 15^13^...^3 mod 16. Notice that 15==-1 (mod 16) which means 15^n=-1 or 1 mod 16 depending on the parity of n. Notice that 13^11^...^3 is always odd. That means 15^13...^3 is -1 mod 16==15 mod 16, so 17^(15) mod 40=33 mod 40 explanation for this step. that means 19^17^...^3==19^33 mod 100, which is 59. explanation for this step That means 19^17^15^13^11^9^7^5^3== 59 (mod 100)
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rchokler
2935 posts
rchokler
#4 h  Saturday at 3:35 PM • 1 Y
Y by Foxellar
$\operatorname{ord}_{100}(19)=10$
$\operatorname{ord}_{10}(17)=\operatorname{ord}_{10}(7)=4$
$\operatorname{ord}_4(15)=\operatorname{ord}_4(3)=2$
$\operatorname{ord}_2(13)=\operatorname{ord}_2(1)=1$

$13^{11^{9^{7^{5^3}}}}\equiv 1\mod 2$
$15^{13^{11^{9^{7^{5^3}}}}}\equiv 15^1\mod 4\equiv 3\mod 4$
$17^{15^{13^{11^{9^{7^{5^3}}}}}}\equiv 17^3\mod 10\equiv 7^3\mod 10\equiv 3\mod 10$
$19^{17^{15^{13^{11^{9^{7^{5^3}}}}}}}\equiv 19^3\mod 100\equiv 59\mod 100$
This post has been edited 1 time. Last edited by rchokler, Saturday at 3:36 PM
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